We present a new approximation algorithm for rate-monotonic multiprocessor scheduling of periodic tasks with implicit deadlines. We prove that for an arbitrary parameter k it yields solutions with at most (3/2 + 1/k)*OPT+9k many processors, thus it gives an asymptotic 3/2-approximation algorithm. This improves over the previously best known ratio of 7/4. Our algorithm can be implemented to run in time O(n^2), where n is the number of tasks. It is based on custom-tailored weights for the tasks such that a greedy maximal matching and subsequent partitioning by a first-fit strategy yields the result.